answer：Okay, so I have this problem where several students are competing in three races. Each race gives points: 5 for first place, 3 for second, and 1 for third. No ties, so each race has distinct placements. The question is asking for the smallest number of points a student needs to earn across the three races to be sure they have more points than any other student. The options are 9, 10, 11, 13, or 15.Hmm, let me break this down. First, I need to figure out the possible points a student can earn. Since there are three races, the maximum points someone can get is 15, which would be winning all three races (5+5+5). But the question isn't about the maximum; it's about the minimum number that guarantees you're ahead of everyone else.So, I guess I need to think about the worst-case scenario. What's the highest someone else could get if I have a certain number of points? I need to find the point total where even if someone else does as well as they can, they can't surpass me.Let me list out the possible point combinations for three races. Each race gives 5, 3, or 1 point, so over three races, the possible totals can range from 3 (1+1+1) up to 15 (5+5+5). But the question is about the minimal total that ensures no one can have more than you.Let me think about the options given: 9, 10, 11, 13, 15. I can probably eliminate 15 because that's the maximum, so you don't need to get 15 to guarantee being ahead. Similarly, 13 is a high number, but maybe not the highest. Let me check.If I get 11 points, what could someone else get? Let's see. If I have 11 points, how did I get there? Maybe 5+5+1, which is 11. Or 5+3+3, which is also 11. So, if I have 11 points, someone else could also have 11 points by getting the other combination. For example, if I got two first places and one third, someone else could get two second places and one first place. Wait, no, because the points are fixed per race. So, if I got two first places, that means two races have already been won by me, so someone else can't get first in those races. Hmm, maybe I need to think about how the points are distributed across races.Let me consider the total points available in all three races. Each race gives out 5+3+1=9 points, so three races give 27 points in total. If I have a certain number of points, the remaining points are distributed among other students.But maybe that's complicating things. Let me think differently. If I want to guarantee that no one can have more points than me, I need to make sure that even if someone else wins as many races as possible, they can't surpass my total.So, if I win two races, that's 5+5=10 points. If I win two races and finish third in the third race, that's 5+5+1=11 points. Alternatively, if I win one race and finish second in two others, that's 5+3+3=11 points. So, 11 points can be achieved in two different ways.But wait, if I have 11 points, could someone else also have 11 points? Let's say I got two first places and one third. Then, in the two races I won, no one else can get first. So, in those two races, the other students can only get second or third. In the third race, I got third, so someone else got first and someone else got second.So, in the two races I won, the maximum someone else can get is 3+3=6 points (second place in both). In the third race, someone else got 5 points for first and someone else got 3 points for second. So, the maximum someone else could get is 5 (from the third race) plus 3+3 from the other two races, which is 11 points. So, someone else could also have 11 points.Therefore, 11 points don't guarantee that I'm ahead of everyone because someone else could also have 11 points.What about 13 points? How can I get 13 points? Let's see: 5+5+3=13. So, I won two races and finished second in the third. Alternatively, 5+3+5=13, same thing. So, if I have 13 points, what's the maximum someone else can get?In the two races I won, no one else can get first. So, in those two races, the maximum someone else can get is 3+3=6 points. In the third race, I got second, so someone else got first (5 points) and someone else got third (1 point). So, the maximum someone else can get is 5 (from the third race) plus 3+3 from the other two races, which is 11 points. So, 11 points is less than 13, so 13 points would guarantee that I'm ahead.Wait, but is there a way for someone else to get more than 11 points if I have 13? Let me check. If I have 13 points, that means I won two races and got second in one. So, in the two races I won, others can only get second or third. In the third race, someone else won, and someone else got third.So, the maximum someone else can get is 5 (from the race I got second) plus 3+3 from the other two races, which is 11. So, 11 is the maximum someone else can get if I have 13. Therefore, 13 points would guarantee that I'm ahead.But wait, what if there are more than two other students? Does that affect anything? Let me think. If there are multiple other students, the points are distributed among them, so the maximum any single student can get is still limited by the points available in each race. So, even if there are multiple students, no one can get more than 11 points if I have 13.Therefore, 13 points seem to be the minimal number that guarantees I have more points than any other student.Wait, but let me check 11 points again. If I have 11 points, someone else could also have 11 points, as I thought earlier. So, 11 doesn't guarantee being ahead. 13 does.What about 10 points? If I have 10 points, how could someone else get more? Let's see. 10 points could be 5+5+0, but wait, you can't get 0 points because in each race, you get at least 1 point for third place. So, the minimum per race is 1 point. So, 10 points could be 5+3+2, but wait, there's no 2 points. The points are only 5, 3, or 1. So, 10 points could be 5+5+0, but that's not possible because you have to get at least 1 point per race. So, 10 points would have to be 5+3+2, but since there's no 2, maybe 5+3+1+1, but that's four races. Wait, no, it's three races. So, 10 points could be 5+3+2, but since there's no 2, it's not possible. Wait, maybe 5+3+1+1, but that's four races. So, actually, 10 points can't be achieved because the points per race are only 5, 3, or 1. So, 10 points would require something like 5+3+2, which isn't possible. So, maybe 10 points isn't a valid total. Wait, no, 5+3+1=9, and 5+5+0=10, but 0 isn't possible. So, 10 points might not be achievable. Hmm, that's confusing.Wait, let me think again. Each race gives 5, 3, or 1 point. So, over three races, the possible totals are combinations of these. So, 5+5+5=15, 5+5+3=13, 5+5+1=11, 5+3+3=11, 5+3+1=9, 3+3+3=9, 3+3+1=7, 3+1+1=5, 1+1+1=3. So, the possible totals are 15, 13, 11, 9, 7, 5, 3. So, 10 points isn't a possible total. So, the options given include 10, but it's not achievable. So, maybe the answer is 11, but as I thought earlier, 11 isn't enough because someone else could also have 11. So, the next possible is 13, which would guarantee being ahead.Wait, but the options include 10, which isn't achievable, so maybe the answer is 11, but since 11 isn't enough, the next is 13. So, the answer should be 13.But let me double-check. If I have 13 points, someone else can have at most 11 points, as I thought earlier. So, 13 is safe. If I have 11 points, someone else could also have 11, so it's not guaranteed. Therefore, the minimal number is 13.Wait, but the options are (A) 9, (B) 10, (C) 11, (D) 13, (E) 15. So, 13 is an option, so that's the answer.But just to be thorough, let me think about 15 points. If I have 15 points, I've won all three races, so no one else can have more than 3 points per race, so the maximum someone else can have is 3+3+3=9, which is much less than 15. So, 15 is definitely enough, but it's not the minimal. The minimal is 13.So, I think the answer is 13, which is option D.

answer：First, I need to determine the length of the shoe in size 8 by working backwards from size 15. Since each unit increase in size corresponds to a 1/5-inch increase in length, each unit decrease in size would correspond to a 1/5-inch decrease in length. The difference in shoe sizes between size 15 and size 8 is 15 - 8 = 7 sizes. Therefore, the total decrease in length from size 15 to size 8 would be 7 sizes * 1/5 inch/size = 7/5 inches = 1.4 inches. Hence, the length of the shoe in size 8 would be 5.9 inches - 1.4 inches = 4.5 inches.Next, I need to find the length of the shoe in size 17. The difference in shoe sizes between size 17 and size 8 is 17 - 8 = 9 sizes. So, the total increase in length from size 8 to size 17 would be 9 sizes * 1/5 inch/size = 9/5 inches = 1.8 inches. Therefore, the length of the shoe in size 17 would be 4.5 inches + 1.8 inches = 6.3 inches.Finally, I can calculate the percentage difference in length between the largest (size 17) and smallest (size 8) sizes of this shoe. The percentage difference is calculated as follows:Percentage difference = [(Larger size - Smaller size) / Smaller size] * 100%Percentage difference = [(6.3 inches - 4.5 inches) / 4.5 inches] * 100%Percentage difference = [1.8 inches / 4.5 inches] * 100%Percentage difference = 0.4 * 100%Percentage difference = 40%So, the largest size of this shoe is 40% longer than the smallest size.

answer：Okay, so I have this problem here about two points in the Cartesian coordinate system. Point A is at (-2, 3) and point B is at (-2, -3). I need to figure out how these two points are positioned relative to each other. The options are symmetry with respect to the x-axis, y-axis, origin, or none of those.Alright, let me start by recalling what symmetry means in the coordinate system. If two points are symmetric with respect to the x-axis, that means if you reflect one point over the x-axis, you get the other point. Similarly, symmetry with respect to the y-axis would mean reflecting over the y-axis, and symmetry with respect to the origin would mean reflecting through the origin, like a 180-degree rotation.Let me write down the coordinates again to visualize them better:- Point A: (-2, 3)- Point B: (-2, -3)Looking at these coordinates, I notice that both points have the same x-coordinate, which is -2. However, their y-coordinates are different: 3 and -3. That seems significant. If I think about reflecting a point over the x-axis, the x-coordinate stays the same, and the y-coordinate changes its sign. So, reflecting point A (-2, 3) over the x-axis would give me (-2, -3), which is exactly point B. That seems to fit the definition of symmetry with respect to the x-axis.Let me check the other options just to be thorough. Symmetry with respect to the y-axis would mean that the y-coordinate stays the same, and the x-coordinate changes its sign. If I reflect point A over the y-axis, I would get (2, 3), which is not point B. So, that doesn't work.Symmetry with respect to the origin would mean both coordinates change their signs. Reflecting point A over the origin would give me (2, -3), which is also not point B. So, that's not it either.Since reflecting over the x-axis gives me point B from point A, and the other symmetries don't apply, I think the correct answer is that they are symmetric with respect to the x-axis.Wait, let me double-check to make sure I didn't make a mistake. If I plot these points on a graph, point A is in the second quadrant because x is negative and y is positive. Point B is in the third quadrant because both x and y are negative. Hmm, actually, if I reflect point A over the x-axis, it should move from the second quadrant to the third quadrant, which is where point B is. That makes sense.But just to be extra sure, let me recall the formal definition of symmetry with respect to the x-axis. A point (x, y) reflected over the x-axis becomes (x, -y). Applying that to point A (-2, 3), we get (-2, -3), which is point B. Perfect, that confirms it.I think I'm confident now that the correct answer is that points A and B are symmetric with respect to the x-axis.

answer：Okay, so I have this problem where I need to find all infinite sequences of positive integers {a_n} that satisfy a certain condition involving a nonnegative integer c. The condition is that for every positive integer n, there are exactly a_n positive integers i such that a_i ≤ a_{n+1} + c. Hmm, that sounds a bit abstract, but let me try to unpack it step by step.First, let me restate the condition to make sure I understand it. For each n, if I look at the value a_{n+1} + c, then the number of terms in the sequence {a_i} that are less than or equal to this value is exactly a_n. So, a_n is counting how many terms in the sequence are ≤ a_{n+1} + c. Interesting.I think it might help to consider some examples or small cases to get a feel for what the sequence could look like. Let me start with c = 0, which is the simplest case since it's a nonnegative integer. If c = 0, the condition becomes: for each n, there are exactly a_n positive integers i such that a_i ≤ a_{n+1}. So, a_n is the number of terms in the sequence that are ≤ a_{n+1}.Wait, that seems recursive. For each n, a_n counts how many terms are ≤ a_{n+1}. So, if I think about the sequence, it's like each term is determined by how many terms before it are less than or equal to the next term. Hmm, that might lead to some kind of increasing sequence.Let me try to construct such a sequence for c = 0. Let's say we start with a_1. Since it's the first term, how many terms are ≤ a_2? Well, a_1 counts how many terms are ≤ a_2. But since we're just starting, a_1 must be 1 because there's only one term so far, which is a_1 itself. So, a_1 = 1.Now, moving on to a_2. a_2 is the number of terms ≤ a_3. But right now, we have a_1 = 1 and a_2 is yet to be determined. So, a_2 must be 2 because there are two terms so far (a_1 and a_2) that are ≤ a_3. Wait, but a_3 is the next term. So, a_2 = 2 implies that a_3 must be such that exactly 2 terms are ≤ a_3. But we already have a_1 = 1 and a_2 = 2, so a_3 must be at least 2. If a_3 is 2, then the number of terms ≤ 2 is 2 (a_1 and a_2), which matches a_2 = 2. So, a_3 = 2.Wait, but then a_3 = 2, so moving on to a_4. a_3 = 2, so a_4 must be the number of terms ≤ a_4. But how does that work? Wait, no, the condition is that for each n, a_n is the number of terms ≤ a_{n+1}. So, for n=3, a_3 = 2 is the number of terms ≤ a_4. So, a_4 must be such that exactly 2 terms are ≤ a_4. But we already have a_1 = 1, a_2 = 2, a_3 = 2. So, if a_4 is 2, then the number of terms ≤ 2 is 3 (a_1, a_2, a_3), which is more than 2. So, a_4 can't be 2. If a_4 is 3, then the number of terms ≤ 3 is 3 (a_1, a_2, a_3), which would mean a_3 = 3, but a_3 is 2. That's a contradiction.Hmm, maybe my initial assumption is wrong. Let me try again. If c = 0, and a_1 = 1, then a_2 must be the number of terms ≤ a_3. But since a_1 = 1, and a_2 is the next term, if a_2 = 1, then a_3 must be such that exactly 1 term is ≤ a_3. But a_1 = 1, so a_3 must be 1. But then a_2 = 1, which would imply that a_3 = 1, and so on. But then the sequence would be all 1s, which might not satisfy the condition for higher n.Wait, let's test that. If all a_n = 1, then for each n, a_n = 1, and a_{n+1} + c = 1 + c. Since c = 0, a_{n+1} + c = 1. So, the number of terms ≤ 1 is infinite, but a_n = 1, which is a contradiction because you can't have exactly 1 term ≤ 1 when all terms are 1. So, that doesn't work.So, maybe c = 0 doesn't allow a constant sequence. Let me try a different approach. Maybe the sequence is strictly increasing? If I assume that the sequence is strictly increasing, then each a_{n+1} is greater than a_n. Then, for each n, a_n counts how many terms are ≤ a_{n+1}. Since the sequence is strictly increasing, the number of terms ≤ a_{n+1} is exactly n+1. So, a_n = n+1. Let me check if that works.If a_n = n+1, then for each n, a_n = n+1, and a_{n+1} + c = (n+2) + c. The number of terms ≤ (n+2) + c is n+2 + c, but a_n = n+1, which is supposed to be equal to the number of terms ≤ a_{n+1} + c. Wait, that doesn't match unless c = 0. But if c = 0, then a_n = n+1, and the number of terms ≤ a_{n+1} is n+2, which is not equal to a_n = n+1. So, that doesn't work either.Hmm, maybe I need to think differently. Let's consider that the sequence might have some constant terms followed by increasing terms. For example, maybe the sequence starts with some number of 1s, then increases. Let me try that.Suppose a_1 = 1, a_2 = 1, a_3 = 2, a_4 = 3, and so on. Let's see if this works for c = 0. For n=1, a_1 = 1, so the number of terms ≤ a_2 = 1 should be 1. But a_1 = 1 and a_2 = 1, so there are 2 terms ≤ 1, which contradicts a_1 = 1. So, that doesn't work.Maybe a_1 = 1, a_2 = 2, a_3 = 2, a_4 = 3, a_5 = 3, etc. Let's see. For n=1, a_1 = 1, so the number of terms ≤ a_2 = 2 should be 1. But a_1 = 1 and a_2 = 2, so there are 2 terms ≤ 2, which contradicts a_1 = 1. Hmm, tricky.Wait, maybe the sequence is such that a_n = n + c + 1. Let me test that. If a_n = n + c + 1, then for each n, a_n = n + c + 1, and a_{n+1} + c = (n + 1 + c + 1) + c = n + 2 + 2c. The number of terms ≤ n + 2 + 2c would be n + 2 + 2c, but a_n = n + c + 1, which is not equal unless c = 0. But even then, it doesn't match. So, that might not be the right approach.Wait, maybe I need to consider that the sequence is such that a_n = n + k for some constant k. Let me assume a_n = n + k and see what k would be. Then, a_{n+1} + c = (n + 1 + k) + c = n + 1 + k + c. The number of terms ≤ n + 1 + k + c is n + 1 + k + c, but a_n = n + k, so we have n + k = n + 1 + k + c, which simplifies to 0 = 1 + c. Since c is nonnegative, this is only possible if c = -1, which is not allowed. So, that approach doesn't work.Hmm, maybe I need to think about the sequence in terms of its cumulative counts. Since a_n is the number of terms ≤ a_{n+1} + c, perhaps the sequence is related to the cumulative distribution of its own terms. That is, a_n is like the inverse of the cumulative distribution function.Wait, let me formalize that. If I define F(x) as the number of terms in the sequence {a_i} that are ≤ x, then the condition is F(a_{n+1} + c) = a_n. So, F is the cumulative count function, and it's related to the sequence itself.If I can express F in terms of the sequence, maybe I can find a recursive relation. Since F(a_{n+1} + c) = a_n, and F is non-decreasing, perhaps F is linear? Or maybe F has a specific form.Wait, if F is linear, say F(x) = kx + b, then F(a_{n+1} + c) = k(a_{n+1} + c) + b = a_n. But I don't know if F is linear. Alternatively, maybe F is piecewise linear or has some other form.Alternatively, perhaps the sequence {a_n} is such that a_{n+1} = F^{-1}(a_n) - c, assuming F is invertible. But I'm not sure if that helps directly.Wait, let's consider that F(a_{n+1} + c) = a_n. So, F is the number of terms ≤ a_{n+1} + c, which is a_n. So, if I think of F as a function that maps x to the number of terms ≤ x, then F(a_{n+1} + c) = a_n. So, F is kind of a step function that increases by 1 at each a_i.Wait, maybe I can model this as F(x) being the number of terms ≤ x, so F(x) = sum_{i=1}^∞ [a_i ≤ x], where [.] is the indicator function. Then, F(a_{n+1} + c) = a_n.Hmm, this seems a bit abstract. Maybe I can think of it in terms of the sequence's growth. If the sequence is increasing, then F(x) would be roughly the position where x appears in the sequence. But I'm not sure.Wait, let's try to consider that the sequence is strictly increasing. If {a_n} is strictly increasing, then F(x) is just the position where x appears in the sequence. So, F(a_{n+1} + c) = n+1. But according to the condition, F(a_{n+1} + c) = a_n. So, a_n = n+1. But earlier, I saw that a_n = n+1 doesn't satisfy the condition because F(a_{n+1} + c) would be n+2 + c, which is not equal to a_n = n+1 unless c = 0, which still doesn't work because n+2 ≠ n+1.Wait, maybe the sequence isn't strictly increasing. Maybe it's non-decreasing. So, some terms can be equal. Let's consider that.Suppose the sequence starts with some number of 1s, then increases. For example, a_1 = 1, a_2 = 1, a_3 = 2, a_4 = 2, a_5 = 3, etc. Let's see if this works for c = 0.For n=1, a_1 = 1, so F(a_2 + 0) = F(1) = 2 (since a_1 and a_2 are 1). But a_1 = 1, which should equal F(a_2) = 2. That's a contradiction.Hmm, maybe the sequence needs to have a different structure. Let me think about the relationship between a_n and a_{n+1}.Given that F(a_{n+1} + c) = a_n, and F is the number of terms ≤ a_{n+1} + c, which is a_n. So, a_n is the count of terms in the sequence up to a certain point.Wait, maybe the sequence is such that a_n = n + c + 1. Let me test this for c = 0. Then, a_n = n + 1. For n=1, a_1 = 2, so F(a_2) = F(3) = 2. But a_2 = 3, so F(3) should be 2. But in reality, F(3) would be the number of terms ≤ 3, which are a_1=2 and a_2=3, so F(3)=2, which matches a_1=2. For n=2, a_2=3, so F(a_3)=F(4)=3. a_3=4, so F(4)=3. But the terms are a_1=2, a_2=3, a_3=4, so F(4)=3, which matches a_2=3. Similarly, for n=3, a_3=4, so F(a_4)=F(5)=4. a_4=5, so F(5)=4, which matches a_3=4. It seems to work for c=0.Wait, so if a_n = n + c + 1, then for c=0, a_n = n + 1, which seems to satisfy the condition. Let me check for c=1. Then, a_n = n + 2. For n=1, a_1=3, so F(a_2 +1)=F(4 +1)=F(5)=3. But a_2=4, so a_2 +1=5. The number of terms ≤5 is a_1=3, a_2=4, a_3=5, so F(5)=3, which matches a_1=3. For n=2, a_2=4, so F(a_3 +1)=F(5 +1)=F(6)=4. a_3=5, so a_3 +1=6. The number of terms ≤6 is a_1=3, a_2=4, a_3=5, a_4=6, so F(6)=4, which matches a_2=4. Similarly, for n=3, a_3=5, so F(a_4 +1)=F(6 +1)=F(7)=5. a_4=6, so a_4 +1=7. The number of terms ≤7 is a_1=3, a_2=4, a_3=5, a_4=6, a_5=7, so F(7)=5, which matches a_3=5. It seems to work for c=1 as well.So, maybe the general solution is a_n = n + c + 1. Let me test this for c=2. Then, a_n = n + 3. For n=1, a_1=4, so F(a_2 +2)=F(5 +2)=F(7)=4. a_2=5, so a_2 +2=7. The number of terms ≤7 is a_1=4, a_2=5, a_3=6, a_4=7, so F(7)=4, which matches a_1=4. For n=2, a_2=5, so F(a_3 +2)=F(6 +2)=F(8)=5. a_3=6, so a_3 +2=8. The number of terms ≤8 is a_1=4, a_2=5, a_3=6, a_4=7, a_5=8, so F(8)=5, which matches a_2=5. It seems consistent.Therefore, it seems that the sequence a_n = n + c + 1 satisfies the given condition for any nonnegative integer c. Let me try to see if there are any other possible sequences or if this is the only one.Suppose there's another sequence that isn't of the form a_n = n + c + 1. Let's say it starts differently. For example, maybe a_1 = c + 2, a_2 = c + 3, and so on, but with some terms repeated. Wait, but if terms are repeated, then F(a_{n+1} + c) would count those repeated terms, which might affect the count. Let me see.Suppose c=0 and a_1=2, a_2=2, a_3=3, a_4=4, etc. For n=1, a_1=2, so F(a_2)=F(2)=2. But a_2=2, so F(2)=2, which is correct because a_1=2 and a_2=2. For n=2, a_2=2, so F(a_3)=F(3)=2. But a_3=3, so F(3)=3, which contradicts a_2=2. So, that doesn't work.Alternatively, maybe a_1=1, a_2=3, a_3=3, a_4=4, etc. For n=1, a_1=1, so F(a_2)=F(3)=1. But a_2=3, so F(3)=2 (a_1=1 and a_2=3), which contradicts a_1=1. So, that doesn't work either.It seems that any deviation from the sequence a_n = n + c + 1 leads to contradictions. Therefore, I think the only possible sequence is a_n = n + c + 1.Wait, let me try to formalize this. Suppose a_n = n + k for some constant k. Then, a_{n+1} + c = (n + 1 + k) + c = n + 1 + k + c. The number of terms ≤ n + 1 + k + c is n + 1 + k + c, but according to the condition, this should be equal to a_n = n + k. So, n + k = n + 1 + k + c, which simplifies to 0 = 1 + c. Since c is nonnegative, this is only possible if c = -1, which is not allowed. So, my earlier assumption that a_n = n + c + 1 must be incorrect in this approach.Wait, but earlier when I tested specific cases, it seemed to work. Maybe I made a mistake in the general approach. Let me re-examine.If a_n = n + c + 1, then a_{n+1} + c = (n + 1 + c + 1) + c = n + 2 + 2c. The number of terms ≤ n + 2 + 2c is n + 2 + 2c, but a_n = n + c + 1. So, n + c + 1 = n + 2 + 2c, which simplifies to c + 1 = 2 + 2c, so -c = 1, which implies c = -1. Again, that's a contradiction because c is nonnegative.Wait, but earlier when I tested specific cases with c=0,1,2, it seemed to work. What's the discrepancy here?Ah, I think I see the mistake. When I assume a_n = n + c + 1, I'm not correctly accounting for the fact that F(a_{n+1} + c) is the number of terms ≤ a_{n+1} + c, which includes all terms up to that point, not just the first n terms. So, my earlier approach was incorrect because I was assuming F(x) = x - k, but that's not the case.Let me try a different approach. Suppose the sequence is a_n = n + c + 1. Then, for each n, a_{n+1} + c = (n + 1 + c + 1) + c = n + 2 + 2c. The number of terms ≤ n + 2 + 2c is the number of terms in the sequence up to that point. Since the sequence is a_n = n + c + 1, the terms are 1 + c + 1, 2 + c + 1, 3 + c + 1, etc. So, the term a_{k} = k + c + 1. Therefore, the number of terms ≤ x is the largest k such that k + c + 1 ≤ x. Solving for k, we get k ≤ x - c - 1. So, the number of terms ≤ x is floor(x - c - 1). Therefore, F(x) = floor(x - c - 1).But according to the condition, F(a_{n+1} + c) = a_n. So, F(a_{n+1} + c) = floor((a_{n+1} + c) - c - 1) = floor(a_{n+1} - 1). Since a_{n+1} = (n + 1) + c + 1 = n + c + 2, then floor(a_{n+1} - 1) = floor(n + c + 2 - 1) = floor(n + c + 1) = n + c + 1. But a_n = n + c + 1, so this matches. Therefore, the condition is satisfied.Wait, that makes sense now. So, even though my initial algebraic approach seemed to lead to a contradiction, when I correctly model F(x) as the number of terms ≤ x, which is floor(x - c - 1), it does satisfy the condition F(a_{n+1} + c) = a_n.Therefore, the sequence a_n = n + c + 1 satisfies the given condition for any nonnegative integer c. I think this is the only solution because any deviation from this sequence would either lead to a contradiction in the count or fail to satisfy the condition for some n.So, putting it all together, the only infinite sequence of positive integers {a_n} that satisfies the condition for all positive integers n is a_n = n + c + 1.

answer：Okay, so I have this problem here about a point P with coordinates (m-1, m+3), and I need to figure out which quadrant it cannot be in. The options are the first, second, third, or fourth quadrant. Hmm, let me think about how to approach this.First, I remember that in the Cartesian coordinate system, the quadrants are divided based on the signs of the x and y coordinates. So, the first quadrant is where both x and y are positive, the second quadrant is where x is negative and y is positive, the third quadrant is where both x and y are negative, and the fourth quadrant is where x is positive and y is negative.So, for point P(m-1, m+3), the x-coordinate is (m-1) and the y-coordinate is (m+3). I need to figure out for which quadrant there's no possible value of m that would make both coordinates satisfy the conditions of that quadrant.Let me start by considering each quadrant one by one.**First Quadrant:**For a point to be in the first quadrant, both x and y must be positive. So, I need:1. m - 1 > 02. m + 3 > 0Let me solve these inequalities:1. m - 1 > 0 ⇒ m > 12. m + 3 > 0 ⇒ m > -3So, for both conditions to be true, m must be greater than 1. That means if m is greater than 1, the point P will be in the first quadrant. So, it's possible for P to be in the first quadrant. Therefore, the first quadrant is not the answer.**Second Quadrant:**For a point to be in the second quadrant, x must be negative and y must be positive. So, I need:1. m - 1 < 02. m + 3 > 0Solving these:1. m - 1 < 0 ⇒ m < 12. m + 3 > 0 ⇒ m > -3So, combining these, m must be between -3 and 1. That means if m is between -3 and 1, the point P will be in the second quadrant. So, it's possible for P to be in the second quadrant. Therefore, the second quadrant is not the answer.**Third Quadrant:**For a point to be in the third quadrant, both x and y must be negative. So, I need:1. m - 1 < 02. m + 3 < 0Solving these:1. m - 1 < 0 ⇒ m < 12. m + 3 < 0 ⇒ m < -3So, combining these, m must be less than -3. That means if m is less than -3, the point P will be in the third quadrant. So, it's possible for P to be in the third quadrant. Therefore, the third quadrant is not the answer.**Fourth Quadrant:**For a point to be in the fourth quadrant, x must be positive and y must be negative. So, I need:1. m - 1 > 02. m + 3 < 0Solving these:1. m - 1 > 0 ⇒ m > 12. m + 3 < 0 ⇒ m < -3Wait a minute, these two conditions seem conflicting. If m has to be greater than 1 and less than -3 at the same time, that's impossible because 1 is greater than -3. There's no value of m that satisfies both m > 1 and m < -3 simultaneously. So, there's no solution for m that would make point P lie in the fourth quadrant.Therefore, the point P(m-1, m+3) cannot be in the fourth quadrant.**Final Answer**boxed{D}

answer：To solve the problem, we start by recognizing that if ( x^2 + 4x + 5 ) is a factor of ( x^4 + Px^2 + Q ), then the polynomial ( x^4 + Px^2 + Q ) can be expressed as the product of ( x^2 + 4x + 5 ) and another quadratic polynomial ( x^2 + bx + c ). We set up the equation:[(x^2 + 4x + 5)(x^2 + bx + c) = x^4 + Px^2 + Q]Expanding the left-hand side, we get:[x^4 + (b + 4)x^3 + (4b + c + 5)x^2 + (5b + 4c)x + 5c]Since the right-hand side is ( x^4 + Px^2 + Q ), we can equate the coefficients of corresponding powers of ( x ) from both sides. This gives us the following system of equations:[begin{cases}b + 4 = 0 4b + c + 5 = P 5b + 4c = 0 5c = Qend{cases}]First, solving ( b + 4 = 0 ) gives ( b = -4 ).Next, substituting ( b = -4 ) into the equation ( 5b + 4c = 0 ), we get:[5(-4) + 4c = 0 implies -20 + 4c = 0 implies 4c = 20 implies c = 5]With ( b = -4 ) and ( c = 5 ), we can now find ( P ) and ( Q ):[P = 4b + c + 5 = 4(-4) + 5 + 5 = -16 + 10 = -6][Q = 5c = 5 times 5 = 25]Finally, adding ( P ) and ( Q ) together:[P + Q = -6 + 25 = 19]